Asymptotic Dimension of Big Mapping Class Groups

2021/10/06 by Curtis Grant, Grant, Curtis, Kasra Rafi +3 · 3 citations
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2110.03087

openalex publication_date 2021/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We show that the big mapping class group of a stable surface of infinite type with a coarsely bounded generating set that contains an essential shift has infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. We also give a topological characterization of essential shifts.

Citations

• Geometry of the mapping class groups III: Quasi-isometric rigidity

Cited by

• Coarse geometry of homeomorphism groups: Classifying countable Stone spaces
• The number of ends of big mapping class groups
• Geometric models and asymptotic dimension for infinite-type surface mapping class groups

Related

• Mapping class groups have finite asymptotic dimension
• Structure and Topological Generation of Big Mapping Class Groups
• Coarsely bounded generating sets for mapping class groups of infinite-type surfaces
• Big mapping class groups are not extremely amenable
• Stable commutator length on big mapping class groups